We introduce a theoretical framework for analyzing federated learning in a generative setting through a teacher-multiple interacting students scenario, in which each student receives a distinct realization of the data, either through a different noise corruption or by accessing a different subset, possibly of varying size. Using theoretical tools in equilibrium disordered system, we analytically show that interactions among students systematically enhance learning performance: highly noisy students require fewer samples to recover the underlying pattern, while low-noise students achieve a larger overlap with the ground-truth signal. We derive the optimal Bayesian conditions for teacher recovery as functions of the sample complexity, noise level, and interaction strength, and validate these predictions through numerical simulations. The resulting dynamics can be mapped onto equilibrium sampling in a Restricted Boltzmann Machine with a structured hidden layer, providing a principled theoretical understanding of how interactions improve distributed generative modeling.

翻译：我们提出一个理论框架，用于分析生成式场景中的联邦学习，该框架通过教师-多交互学生场景实现，其中每个学生接收独特的数据实现，这些数据或通过不同噪声破坏，或通过访问不同子集（可能大小各异）获得。利用平衡无序系统的理论工具，我们解析地证明学生间的交互系统性地提升学习性能：高噪声学生需更少样本恢复潜在模式，而低噪声学生与真实信号的重叠度更大。我们推导出教师恢复的最优贝叶斯条件（作为样本复杂度、噪声水平和交互强度的函数），并通过数值模拟验证这些预测。由此产生的动力学可映射为具有结构化隐藏层的受限玻尔兹曼机中的平衡采样，为理解交互如何改进分布式生成建模提供理论基础。